Convergent matrix

Background

When successive powers of a matrix T become small (that is, when all of the entries of T approach zero, upon raising T to successive powers), the matrix T converges. A regular splitting of a non-singular matrix A results in a convergent matrix T. A semi-convergent splitting of a matrix A results in a semi-convergent matrix T. A general iterative method converges for every initial vector if T is convergent, and under certain conditions if T is semi-convergent.

Definition

We call an n × n matrix T a convergent matrix if

for each i = 1, 2, ..., n and j = 1, 2, ..., n.

Example

Let

Computing successive powers of T, we obtain

and, in general,

Since

and

T is a convergent matrix. Note that ρ(T) = 1/4, where ρ(T) represents the spectral radius of T, since 1/4 is the only eigenvalue of T.

Characterizations

Let T be an n × n matrix. The following properties are equivalent to T being a convergent matrix:

1. lim k → ∞ ∥ T k ∥ = 0 , for some natural norm;
2. lim k → ∞ ∥ T k ∥ = 0 , for all natural norms;
3. ρ ( T ) < 1 ;
4. lim k → ∞ T k x = 0 , for every x.

Iterative methods

A general iterative method involves a process that converts the system of linear equations

into an equivalent system of the form

for some matrix T and vector c. After the initial vector x⁽⁰⁾ is selected, the sequence of approximate solution vectors is generated by computing

for each k ≥ 0. For any initial vector x⁽⁰⁾ ∈ R n , the sequence { x ( k ) } k = 0 ∞ defined by (4), for each k ≥ 0 and c ≠ 0, converges to the unique solution of (3) if and only if ρ(T) < 1, that is, T is a convergent matrix.

Regular splitting

A matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. In the system of linear equations (2) above, with A non-singular, the matrix A can be split, that is, written as a difference

so that (2) can be re-written as (4) above. The expression (5) is a regular splitting of A if and only if B⁻¹ ≥ 0 and C ≥ 0, that is, B⁻¹ and C have only nonnegative entries. If the splitting (5) is a regular splitting of the matrix A and A⁻¹ ≥ 0, then ρ(T) < 1 and T is a convergent matrix. Hence the method (4) converges.

Semi-convergent matrix

We call an n × n matrix T a semi-convergent matrix if the limit

exists. If A is possibly singular but (2) is consistent, that is, b is in the range of A, then the sequence defined by (4) converges to a solution to (2) for every x⁽⁰⁾ ∈ R n if and only if T is semi-convergent. In this case, the splitting (5) is called a semi-convergent splitting of A.